# Chapter 12: The Atomic Nuleus¶

## Example 12.1, Page 432¶

In [1]:
import math

#Variable declaration
h = 6.62e-034;    # Planck's constant, Js
h_bar = h/(2*math.pi);    # Reduced Planck's constant, Js
m = 1.67e-027;    # Rest mass of proton, kg
e = 1.602e-019;    # Energy equivalent of 1 eV, J
c = 3.00e+008;    # Speed of light, m/s
delta_x = 8.0e-015;    # Size of the nucleus, m

#Calculations
delta_p = h_bar/(2*delta_x*e);    # Uncertainty in momentum of proton from Heisenberg Uncertainty Principle, eV-s/m
p_min = delta_p;    # Minimum momentum of the proton inside the nucleus, eV-s/m
K = (p_min*c)**2*e/(2*m*c**2*1e+006);    # The minimum kinetic energy of the proton in a medium sized nucleus, MeV

#Result
print "The minimum kinetic energy of the proton in a medium sized nucleus = %4.2f MeV"%K

The minimum kinetic energy of the proton in a medium sized nucleus = 0.08 MeV


## Example 12.2, Page 436¶

In [3]:
import math

#Variable declaration
c = 3.00e+008;    # Speed of light, m/s
e = 1.602e-019;    # Energy equivalent of 1 eV, J
m_e = 0.511;    # Rest mass energy of electron, MeV
m_p = 938.3;    # Rest mass energy of proton, MeV
h = 6.62e-034;    # Planck's constant, Js
A = 40;    # Mass number of Ca-40
r0 = 1.2;    # Nuclear radius constant, fm

#Calculations&Results
R = r0*A**(1./3);    # Radius of Ca-40 nucleus, fm
print "The radius of Ca-40 nucleus = %3.1f fm"%R
lamda = 2.0;    # de-Broglie wavelength to distinguish a distance at least half the radius, fm
# Electron energy
E = math.ceil(math.sqrt(m_e**2+(h*c/(lamda*e*1e+006*1e-015))**2));    # Total energy of the probing electron, MeV
K = E - m_e;    # Kinetic energy of probing electron, MeV
print "The kinetic energy of probing electron = %3d MeV"%math.ceil(K)
# Proton energy
E = math.ceil(math.sqrt(m_p**2+(h*c/(lamda*e*1e+006*1e-015))**2));    # Total energy of the probing electron, MeV
K = E - m_p;    # Kinetic energy of probing electron, MeV
print "The kinetic energy of probing proton = %3d MeV"%math.ceil(K)

#answers differ due to rounding errors

The radius of Ca-40 nucleus = 4.1 fm
The kinetic energy of probing electron = 620 MeV
The kinetic energy of probing proton = 187 MeV


## Example 12.3, Page 437¶

In [4]:
#Variable declaration
A_U238 = 238;    # Mass number of U-238
A_He4 = 4;    # Mass number of He-4
r0 = 1.2;    # Nuclear radius constant, nm

#Calculations&Results
R_U238 = r0*A_U238**(1./3);    # Radius of U-238 nucleus, fm
R_He4 = r0*A_He4**(1./3);    # Radius of He-4 nucleus, fm
print "The radii of U-238 and He-4 nuclei are %3.1f fm and %3.1f fm repectively"%(R_U238, R_He4)
ratio = R_U238/R_He4;    # Ratio of the two radii
print "The ratio of radius to U-238 to that of He-4 = %3.1f"%ratio

The radii of U-238 and He-4 nuclei are 7.4 fm and 1.9 fm repectively
The ratio of radius to U-238 to that of He-4 = 3.9


## Example 12.4, Page 438¶

In [5]:
#Variable declaration
h = 6.62e-034;    # Planck's constant, Js
c = 3.00e+008;    # Speed of light, m/s
e = 1.602e-019;    # Energy equivalent of 1 eV, J
B = 2.0;    # Applied magnetic field, T
mu_N = 3.15e-008;    # Nucleon magnetic moment, eV/T

#Calculations
mu_p = 2.79*mu_N;    # Proton magnetic moment, eV/T
delta_E = 2*mu_p*B;    # Energy difference between the up and down proton states, eV
f = delta_E*e/h;    # Frequency of electromagnetic radiation that flips the proton spins, Hz
lamda = c/f;    # Wavelength of electromagnetic radiation that flips the proton spins, m

#Results
print "The energy difference between the up and down proton states = %3.1e eV"%delta_E
print "The frequency of electromagnetic radiation that flips the proton spins = %2d MHz"%(f/1e+006)
print "The wavelength of electromagnetic radiation that flips the proton spins = %3.1f m"%lamda

The energy difference between the up and down proton states = 3.5e-07 eV
The frequency of electromagnetic radiation that flips the proton spins = 85 MHz
The wavelength of electromagnetic radiation that flips the proton spins = 3.5 m


## Example 12.5, Page 443¶

In [6]:
#Variable declaration
c = 3.00e+008;    # Speed of light, m/s
e = 1.602e-019;    # Energy equivalent of 1 eV, J
u = 931.5;    # Energy equivalent of 1 amu, MeV
m_n = 1.008665;    # Mass of a neutron, u
M_H = 1.007825;    # Mass of a proton, u
M_He = 4.002603;    # Mass of helium nucleus
M_Be = 8.005305;    # Mass of Be-8, u

#Calculations&Results
B_Be = (4*m_n+4*M_H - M_Be)*u;
B_Be_2alpha = (2*M_He - M_Be)*u;
print "The binding energy of Be-8 = %4.1f MeV and is positive"%B_Be
if (B_Be_2alpha > 0):
print "The Be-4 is stable w.r.t. decay into two alpha particles.";
else:
print "The Be-4 is unstable w.r.t. decay into two alpha particles since the decay has binding energy of %5.3f MeV"%B_Be_2alpha

The binding energy of Be-8 = 56.5 MeV and is positive
The Be-4 is unstable w.r.t. decay into two alpha particles since the decay has binding energy of -0.092 MeV


## Example 12.6, Page 444¶

In [7]:
#Variable declaration
Z = 92;    # Atomic number of U-238
A = 238;    # Mass number of U-238

#Calculations
E_Coul = 0.72*Z*(Z-1)*A**(-1./3);    # Total Coulomb energy of U-238, MeV

#Result
print "The total Coulomb energy of U-238 = %3d MeV"%E_Coul

The total Coulomb energy of U-238 = 972 MeV


## Example 12.8, Page 447¶

In [8]:
#Variable declaration
u = 931.5;    # Energy equivalent of 1 amu, MeV
m_p = 1.007825;    # Mass of proton, amu
m_n = 1.008665;     # Mass of neutron, amu
M_Ne = 19.992440; # Mass of Ne-20 nucleus, amu
M_Fe = 55.934942;    # Mass of Fe-56 nucleus, amu
M_U = 238.050783;    # Mass of U-238 nucleus, amu
A_Ne = 20;    # Mass number of Ne-20 nucleus
A_Fe = 56;    # Mass number of Fe-56 nucleus
A_U = 238;    # Mass number of U-238 nucleus

#Calculations
BE_Ne = (10*m_p+10*m_n-M_Ne)*u;    # Binding energy of Ne-20 nucleus, MeV
BE_Fe = (26*m_p+30*m_n-M_Fe)*u;    # Binding energy of Fe-56 nucleus, MeV
BE_U = (92*m_p+146*m_n-M_U)*u;    # Binding energy of U-238 nucleus, MeV

#Results
print "The binding energy per nucleon for Ne-20 : %4.2f MeV/nucleon"%(BE_Ne/A_Ne)
print "The binding energy per nucleon for Fe-56 : %4.2f MeV/nucleon"%(BE_Fe/A_Fe)
print "The binding energy per nucleon for U-238 : %4.2f MeV/nucleon"%(BE_U/A_U)

The binding energy per nucleon for Ne-20 : 8.03 MeV/nucleon
The binding energy per nucleon for Fe-56 : 8.79 MeV/nucleon
The binding energy per nucleon for U-238 : 7.57 MeV/nucleon


## Example 12.10, Page 451¶

In [9]:
import math

#Variable declaration
N_A = 6.023e+023;    # Avogadro's number
T = 138*24*3600;    # Half life of Po-210, s
R = 2000;    # Activity of Po-210, disintegrations/s
M = 0.210;    # Gram molecular mass of Po-210, kg

#Calculations
f = 1/3.7e+010;    # Conversion factor to convert from decays/s to Ci
A = R*f/1e-006;    # Activity of Po-210, micro-Ci
N = R*T/math.log(2);    # Number of radioactive nuclei of Po-210, nuclei
m = N*M/N_A;    # Mass of Po-210 sample, kg

#Results
print "The activity of Po-210 = %5.3f micro-Ci"%A
print "The mass of Po-210 sample = %3.1e kg"%m

The activity of Po-210 = 0.054 micro-Ci
The mass of Po-210 sample = 1.2e-14 kg


## Example 12.11, Page 452¶

In [10]:
import math

#Variable declaration
T = 110;    # Half life of F-18, min
f_remain = 0.01;    # Fraction of the F-18 sample remained

#Calculations
t = -math.log(0.01)/(math.log(2)*60)*T;    # Time taken by the F-18 sample to decay to 1 percent of its initial value, h

#Result
print "The time taken for 99 percent of the F-18 sample to decay = %4.1f h"%t

The time taken for 99 percent of the F-18 sample to decay = 12.2 h


## Example 12.12, Page 452¶

In [11]:
import math

#Variable declaration
N_A = 6.023e+023;    # Avogadro's number
M = 10e+03;    # Mass of the U-235, g
M_U235 = 235;    # Molecular mass of U-235, g
t_half = 7.04e+008;    # Half life of U-235, y

#Calculations
N = M*N_A/M_U235;    # Number of U-235 atoms in 10 kg sample
R = math.log(2)*N/t_half;    # The alpha activity of 10 kg sample of U-235, decays/y

#Result
print "The alpha activity of 10 kg sample of U-235 = %3.1e Bq"%(R/(365.25*24*60*60))

The alpha activity of 10 kg sample of U-235 = 8.0e+08 Bq


## Example 12.13, Page 453¶

In [14]:
#Variable declaration
u = 931.5;    # Energy equivalent of 1 u, MeV
M_U230 = 230.033927;    # Atomic mass of U-230, u
m_n = 1.008665;    # Mass of a neutron, u
M_H = 1.007825;    # Mass of hydrogen, u
M_U229 = 229.033496;    # Gram atomic mass of U-230

#Calculations&Results
Q = (M_U230 - M_U229 - m_n)*u;    # Q-value of the reaction emitting a neutron
if (Q < 0):
print "The neutron decay in this reaction is not possible."
else:
print "The neutron decay in this reaction is possible."

Q = (M_U230 - M_U229 - M_H)*u;    # Q-value of the reaction emitting a proton
if (Q < 0):
print "The proton decay in this reaction is not possible."
else:
print "The proton decay in this reaction is possible."

The neutron decay in this reaction is not possible.
The proton decay in this reaction is not possible.


## Example 12.15, Page 461¶

In [13]:
#Variable declaration
u = 931.5;    # Energy equivalent of 1 u, MeV
M_Fe55 = 54.938298;    # Atomic mass of Fe-55, u
M_Mn55 = 54.938050;    # Atomic mass of Mn-55, u
m_e = 0.000549;    # Mass of the electron, u

#Calculations&Results
Q = (M_Fe55 - M_Mn55 - 2*m_e)*u;    # Q-value of the reaction undergoing beta+ decay, MeV
if (Q < 0):
print "The beta+ decay is not allowed for Fe-55"
else:
print "The beta+ decay is allowed for Fe-55"

Q = (M_Fe55 - M_Mn55)*u;    # Q-value of the reaction undergoing electron capture, MeV
if (Q < 0):
print "Fe-55 may not undergo electron capture"
else:
print "Fe-55 may undergo electron capture"

The beta+ decay is not allowed for Fe-55
Fe-55 may undergo electron capture


## Example 12.16, Page 462¶

In [15]:
#Variable declaration
def check_allowance(Q, decay_type):
if (Q < 0):
print "The %s is not allowed for Ac-226"%decay_type
else:
print "The %s is allowed for Ac-226"%decay_type

u = 931.5;    # Energy equivalent of 1 u, MeV
M_Ac226 = 226.026090;    # Atomic mass of Ac-226, u
M_Fr222 = 222.017544;    # Atomic mass of Fr-222, u
M_He4 = 4.002603;    # Atomic mass of He-4, u
M_Th226 = 226.024891;    # Atomic mass of M_Th226, u
M_Ra226 = 226.025403;    # Atomic mass of M_Ra226, u
m_e = 0.000549;    # Mass of the electron, u

#Calculations
# Alpha Decay
Q = (M_Ac226 - M_Fr222 - M_He4)*u;    # Q-value of the reaction undergoing alpha decay, MeV
check_allowance(Q, "alpha decay");
# Beta- Decay
Q = (M_Ac226 - M_Th226)*u;    # Q-value of the reaction undergoing beta- decay, MeV
check_allowance(Q, "beta- decay");
# Beta+ Decay
Q = (M_Ac226 - M_Ra226 - 2*m_e)*u;    # Q-value of the reaction undergoing beta+ decay, MeV
check_allowance(Q, "beta+ decay");
# Electron Capture
Q = (M_Ac226 - M_Ra226)*u;    # Q-value of the reaction undergoing electron capture, MeV
check_allowance(Q, "electron capture");

The alpha decay is allowed for Ac-226
The beta- decay is allowed for Ac-226
The beta+ decay is not allowed for Ac-226
The electron capture is allowed for Ac-226


## Example 12.17, Page 463¶

In [16]:
#Variable declaration
u = 931.5;    # Energy equivalent of 1 u, MeV
E_ex = 0.072;    # Energy of the excited state, MeV

#Calculations
M = 226*u*1e+0;    # Energy equivalent of atomic mass of Th-226, MeV
x = E_ex/(2*M);    # The error introduced in the gamma ray energy by approximation

#Result
print "The error introduced in the gamma ray energy by approximation = %3.1e"%x

The error introduced in the gamma ray energy by approximation = 1.7e-07


## Example 12.18, Page 467¶

In [17]:
import math

#Variable declaration
t_half = 4.47e+009;    # The half life of uranium ore, years
R_prime = 0.60;    # The ratio of Pb206 abundance to that of U238

#Calculations
t = t_half/math.log(2)*math.log(R_prime + 1);    # Age of the uranuim ore, years

#Result
print "The age of U-238 ore = %3.1e years"%t

The age of U-238 ore = 3.0e+09 years


## Example 12.19, Page 469¶

In [18]:
import math

#Variable declaration
t_half = 5730;    # The half life of uranium ore, years
R0 = 1.2e-012;    # The initial ratio of C14 to C12 at the time of death
R = 1.10e-012;    # The final ratio of C14 to C12 t years after death

#Calculations
# As R = R0*math.exp(-lambda*t), solving for t
t = -math.log(R/R0)*t_half/math.log(2);    # Age of the bone, years

#Result
print "The %3d years age of bone does not date from the Roman Empire."%math.ceil(t)

The 720 years age of bone does not date from the Roman Empire.